A 1-dimensional face is called an edge, and consists of a line segment.

* 1-face – any 1-dimensional edge

The link of a simplex s in a simplicial complex K is a subcomplex of K consisting of the simplices t that are disjoint from s and such that both s and t are faces of some higher-dimensional simplex in K. For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices, edges, and triangles, the link of a vertex s consists of the cycle of vertices and edges surrounding s: if t is a vertex in this cycle, it and s are both endpoints of an edge of K, and if t is an edge in this cycle, it and s are both faces of a triangle of K. This cycle is homeomorphic to a circle, which is a 1-dimensional sphere.

The square is bounded by 1-dimensional lines, the cube by 2-dimensional areas, and the tesseract by 3-dimensional volumes.

For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but one can make do with a single coordinate ( the polar coordinate angle ), so the circle is 1-dimensional even though it exists in the 2-dimensional plane.

A regular line, for instance, is conventionally understood to be 1-dimensional ; if such a curve is divided into pieces each 1 / 3 the length of the original, there are always 3 equal pieces.

It is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured.

In even simpler terms, one can consider that points can be thought of as the boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds.

A vector can be represented as a 1-dimensional array and is a 1st-order tensor.

For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell x < sub > i </ sub >< sup > t </ sup >— where t is the time step ( vertical ), and i is the index ( horizontal ) in one generation — is

In some very specific cases, the detection of the phosphorylation as a shift in the protein's electrophoretic mobility is possible on simple 1-dimensional SDS-PAGE gels, as it's described for instance for a transcriptional coactivator by Kovacs et al.

We now use the assumption that is large compared to other scales in the problem ; we therefore neglect the last term in the equation, and get a 1-dimensional diffusion equation:

IBM's recent work on racetrack memory is essentially a 1-dimensional version of bubble, bearing an even closer relationship to the original serial twistor concept.

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R < sup > 3 </ sup > which is called the axis of rotation ( this is Euler's rotation theorem ).

Since the center of U ( n ) is a 1-dimensional abelian normal subgroup of U ( n ), the unitary group is not semisimple.

These wells present only a 1-dimensional segment through the Earth and the skill of inferring 3-dimensional characteristics from them is one of the most fundamental in petroleum geology.

If the assumptions used to justify this simplified approximation ( i. e. steady-state heat conduction, no convection or advection ) are accepted, we define the simple 1-dimensional heat diffusion equation where temperature T at a depth z and time t is given by the equation:

Although that is strictly speaking a question about a real vector bundle ( the " hairs " on a ball are actually copies of the real line ), there are generalizations in which the hairs are complex ( see the example of the complex hairy ball theorem below ), or for 1-dimensional projective spaces over many other fields.

In 1-dimensional quantum mechanics, instantons describe tunneling, which is invisible in perturbation theory.

Cosmic strings are hypothetical 1-dimensional ( spatially ) topological defects which may have formed during a symmetry breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking is not simply connected.

* A graph is a 1-dimensional CW-complex.

Suppose that F is a ( 1-dimensional ) formal group law over R. Its formal group ring ( also called its hyperalgebra or its covariant bialgebra ) is a cocommutative Hopf algebra H constructed as follows.

Electrons behave as beams of energy, and in the presence of a potential U ( z ), assuming 1-dimensional case, the energy levels ψ < sub > n </ sub >( z ) of the electrons are given by solutions to Schrödinger ’ s equation,

So, just like one can find the value of an Integral ( f = dF ) over a 1-dimensional manifolds () by considering the anti-derivative ( F ) at the 0-dimensional boundaries (), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals ( dω ) over n-dimensional manifolds ( Ω ) by considering the anti-derivative ( ω ) at the ( n-1 )- dimensional boundaries ( dΩ ) of the manifold.

For example, the pencil of curves ( 1-dimensional linear system of conics ) defined by is non-degenerate for but is degenerate for concretely, it is an ellipse for two parallel lines for and a hyperbola with < math > a < 0 </ math > – throughout, one axis has length 2 and the other has length which is infinity for

For example, adding an extra tape to the Turing machine, giving it a 2-dimensional ( or 3 or any-dimensional ) infinite surface to work with can all be simulated by a Turing machine with the basic 1-dimensional tape.

in n-dimensions at an arbitrary k, with respect to k. The volume, area or length in 3, 2 or 1-dimensional k-spaces are expressed by

For longitudinal phonons in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional k-space, as shown in Figure 2, is given by

It was invented in 1992 by Steven R. White and it is nowadays the most efficient method for 1-dimensional systems.

This linear transformation, when applied to C ( as a representation of the second copy of G × G ), would give as its image the 1-dimensional subrepresentation generated by

where M is m-dimensional, W is m + 1-dimensional, is diffeomorphic to and is obtained from by the attachment of i-handles.

A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg – de Vries equation ( which describes 1-dimensional non-dissipative fluid dynamics in shallow basins ), could be understood by viewing these equations as infinite dimensional integrable

An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.

* Phase line, 1-dimensional case

Recent research predicts analytically that it cannot exceed a density limit of 63. 4 % This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres ( i. e. line segments or disks ) will yield a regular packing.

The BCS of a line segment ( 1-simplex ) consists of two smaller segments, each connecting one endpoint ( 0-dimensional face ) of to the midpoint of itself ( 1-dimensional face ).

Rather than viewing a 1-dimensional signal ( a function, real or complex-valued, whose domain is the real line ) and some transform ( another function whose domain is the real line, obtained from the original via some transform ), time – frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time – frequency transform.

This last integrand can be recognized as the derivative of the earlier integrand ( with respect to r ) shows that a planimeter computes the area integral in terms of the derivative, which is reflected in Green's theorem, which equates a line integral of a function on a ( 1-dimensional ) contour to the ( 2-dimensional ) integral of the derivative.

A sphere formed using the Chebyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using Manhattan distance is an octahedron: these are dual polyhedra, but among cubes, only the square ( and 1-dimensional line segment ) are self-dual polyhedra.

* Phase line, 1-dimensional case

Digital cameras use a 1-dimensional array sensor to take 1-pixel-wide sequential images of the finish line.